Cascade Framework: Deriving Standard Model Observables from x³ = x² + 1 (v6)

The Cascade Framework derives the complete Standard Model of particle physics and its cosmological constants from a single cubic polynomial, x³ = x² + 1, with zero free parameters. The polynomial’s real root ψₛ ≈ 1. 4656 and its two complex conjugate roots — whose argument θ₃ ≈ 106. 4° is the Berry phase angle of the orbit — generate 28 independent observables at sub-0. 1% accuracy, including all electroweak and CKM parameters, all fermion mass ratios, Newton’s gravitational constant, the dark energy fraction Ω_Λ = 1/ψₛ exactly, and the neutrino mass-squared ratio. The polynomial’s negative prime discriminant (−31) mandates complex conjugate roots, which in turn force a quasi-closure orbit, a quasi-closure comma C, and the forward arrow of time — making the framework’s temporal structure algebraically unavoidable rather than chosen. Version 6 presents eight papers, 45 proved exact algebraic theorems, and the framework’s first external experimental confirmation. Version 6 introduces four major layers of new mathematics. The CM-theoretic structure of x³ = x² + 1 is completed through Theorems T39–T46: the Hilbert class field H = Q (ψₛ, √−31) is confirmed to 200 decimal places, the Stark unit cube ε³ is derived in exact closed form with the Shimura multiplier identity, and a cascade polynomial family is identified in which every member carries discriminant −31·m², establishing the prime 31 as the universal arithmetic signature of the polynomial. Theorem T48 derives N₀ = 55 uniquely from three cascade-internal constraints with no reference to observational data, making the 6/6 DESI DR2 Hubble parameter sign-match a genuine a priori prediction; six falsifiable H₀ recovery redshifts where H (z) = H_ΛCDM exactly are predicted and developed in the standalone cosmological dynamics paper (3b). The Q (s³) Quasi-Spectral Stability Structure is introduced as a new mathematical object unifying the cascade’s identities as quasi-crystal, arithmetic motive, thermodynamic system, and noncommutative geometric space, with dark matter and the arrow of time landing exactly on the Bridgeland amplitude/phase wall, and Conjecture C49 connecting that wall to the critical line Re (s) = 1/2. Finally, a harmonic analysis layer identifies the cascade as a BIBO-stable IIR filter with dark energy as its gain margin, the quasi-closure comma as the Mellin residue of the H (z) oscillation, the Riemann zeros as shadow orbits at gain margin exactly 1, and the Sturmian sequence as the most thermodynamically stable entropy distribution possible for the 13-station orbit. The result is a framework that can now be understood at five simultaneously valid levels: as a number-theoretic object inside the Hilbert class field of Q (√−31), as a Langlands-dual pair whose modular form 31. 1. b. a encodes the amplitude side of the same physics the orbit encodes on the phase side, as a quasi-crystalline torus knot T (44, 13) whose Fibonacci-gap Alexander polynomial and Jones invariant confirm the algebraic structure topologically, as a cosmological model making specific falsifiable predictions about the Hubble parameter oscillation testable by DESI DR3 and Euclid, and as a filter-theoretic stability system whose dark energy parameter places the universe at the precise gain margin that allows resonance without divergence. The collection is organized as Papers 1 through 7 plus the standalone cosmological dynamics paper (3b) ; the computation log previously published separately has been retired and its numerical verifications integrated into Paper 3b’s appendix.